We investigate a compressive sensing framework in which the sensors introduce a distortion to the measurements in the form of unknown gains. We focus on blind calibration, using measures performed on multiple unknown (but sparse) signals and formulate the joint recovery of the gains and the sparse signals as a convex optimization problem. The first proposed approach is an extension to the basis pursuit optimization which can estimate the unknown gains along with the unknown sparse signals. Demonstrating that this approach is successful for a sufficient number of input signals except in cases where the phase shifts among the unknown gains varies significantly, a second approach is proposed that makes use of quadratic basis pursuit optimization to calibrate for constant amplitude gains with maximum variance in the phases. An alternative form of this approach is also formulated to reduce the complexity and memory requirements and provide scalability with respect to the number of input signals. Finally a third approach is formulated which combines the first two approaches for calibration of systems with any variation in the gains. The performance of the proposed algorithms are investigated extensively through numerical simulations, which demonstrate that simultaneous signal recovery and calibration is possible when sufficiently many (unknown, but sparse) calibrating signals are provided.